The radical well-mixed closure of a difference (sigma-)ideal is conjectured to preserve binomiality: for every binomial sigma-ideal II of a difference polynomial ring K{y1,…,yn}K\{y_1,\dots,y_n\} over a characteristic-zero difference field, is the smallest sigma-ideal containing II that is both radical and well-mixed (well-mixed meaning fg∈J⇒fσ(g)∈Jfg\in J\Rightarrow f\sigma(g)\in J) again generated by binomials? The question completes the closure calculus for ordinary binomial difference ideals initiated by Gao, Huang and Yuan (2017), who settled the Laurent case in full and proved the reflexive and perfect closures binomial in the ordinary case while leaving the well-mixed closure open. Wang then showed by example that the plain well-mixed closure of a binomial sigma-ideal can fail to be binomial — the answer depends on the action of σ\sigma on the base field — and posed the radical well-mixed closure version above as an open conjecture (Problem 3.21 in the journal version, Problem 3.18 in the 2016 preprint), which remains unresolved. The monomial analogue of the conjecture is known to hold.

Background

Difference algebra studies algebraic difference equations — systems relating an unknown to its transform under a distinguished operator — by the methods of commutative algebra. A difference field (sigma-field) is a field KK equipped with a field endomorphism σ\sigma; throughout, KK has characteristic zero. For indeterminates y1,…,yny_1,\dots,y_n, the difference polynomial ring (sigma-polynomial ring) is K{y1,…,yn}=K[yj(k):1≤j≤n, k≥0]K\{y_1,\dots,y_n\}=K[y_j^{(k)}:1\le j\le n,\ k\ge0], where yj(k)y_j^{(k)} stands for σk(yj)\sigma^k(y_j) and σ\sigma acts by shifting kk to k+1k+1. A sigma-ideal is an ordinary ideal of this ring that is closed under σ\sigma. Several ideal classes are specific to difference algebra and have no purely algebraic counterpart: a sigma-ideal II is reflexive if σ(a)∈I\sigma(a)\in I implies a∈Ia\in I; well-mixed if fg∈Ifg\in I implies fσ(g)∈If\sigma(g)\in I (well-mixed ideals arise, for instance, in the difference Nullstellensatz and in the theory of difference schemes); and perfect if whenever a product of transforms of a polynomial pp lies in II, then p∈Ip\in I — the difference-algebraic analogue of a radical ideal. Every perfect sigma-ideal is well-mixed and radical, and every sigma-prime ideal is perfect. Each class defines a closure operation on sigma-ideals: the well-mixed closure ⟨I⟩\langle I\rangle (the smallest well-mixed sigma-ideal containing II), the radical well-mixed closure ⟨I⟩r\langle I\rangle_r (the smallest sigma-ideal containing II that is both radical and well-mixed), and the perfect closure {I}\{I\}, with ⟨I⟩⊆⟨I⟩r⊆{I}\langle I\rangle\subseteq\langle I\rangle_r\subseteq\{I\}.

By a sigma-binomial one means a sigma-polynomial with at most two terms, aM1+bM2aM_1+bM_2 with a,b∈Ka,b\in K and M1,M2M_1,M_2 sigma-monomials (products of powers of transforms of the yjy_j); a binomial sigma-ideal is a sigma-ideal generated, possibly by infinitely many, sigma-binomials. In commutative algebra, Eisenbud and Sturmfels proved that binomial ideals are remarkably stable: the radical of a binomial ideal is binomial and binomial ideals admit binomial primary decompositions. Gao, Huang and Yuan (2017) initiated the difference analogue of this theory. In the Laurent sigma-ring, where all sigma-monomials are invertible, they established a dictionary between Laurent binomial sigma-ideals and lattices over Z[x]\mathbb{Z}[x] (the ring of exponents of transforms) and used it to prove that the reflexive, well-mixed, and perfect closures of a Laurent binomial sigma-ideal are again binomial, with support lattices given by the corresponding saturations. For ordinary (non-Laurent) binomial sigma-ideals in K{y1,…,yn}K\{y_1,\dots,y_n\} they proved that the reflexive closure is binomial and — over an algebraically closed inversive base field — that the perfect closure is binomial, and they explicitly left one case open: whether the well-mixed closure of a binomial sigma-ideal is binomial.

That open case was settled negatively in general by Wang (2016 preprint, published in Communications in Algebra 2018): for the sigma-ideal generated by y12(y3−y4)y_1^2(y_3-y_4) and y22(y3−y4)y_2^2(y_3-y_4) in C{y1,y2,y3,y4}\mathbb{C}\{y_1,y_2,y_3,y_4\}, the well-mixed closure is not a binomial sigma-ideal when σ\sigma is the identity on C\mathbb{C}, whereas the same well-mixed closure is binomial when σ\sigma is complex conjugation — binomiality of the well-mixed closure genuinely depends on the action of σ\sigma on the base field. In the same paper Wang posed the surviving question (Problem 3.21): the radical well-mixed closure, sitting between the possibly non-binomial well-mixed closure and the perfect closure (which is binomial over an algebraically closed inversive base field), is conjectured always to be binomial. The monomial analogue is known: Wang proved that the well-mixed closure of a monomial sigma-ideal need not be monomial, but its radical well-mixed closure always is, which is the main structural evidence for the conjecture.

Problem Statement

Let KK be a difference field of characteristic zero with difference operator σ\sigma, let K{y1,…,yn}K\{y_1,\dots,y_n\} be the sigma-polynomial ring in y1,…,yny_1,\dots,y_n, and let II be a binomial sigma-ideal of K{y1,…,yn}K\{y_1,\dots,y_n\}, that is, a sigma-ideal generated by sigma-polynomials with at most two terms. Is the radical well-mixed closure ⟨I⟩r\langle I\rangle_r of II — the smallest sigma-ideal containing II that is both radical and well-mixed, where well-mixed means fg∈Jfg\in J implies fσ(g)∈Jf\sigma(g)\in J — itself a binomial sigma-ideal?

This is Problem 3.21 (Problem 3.18 in the 2016 preprint version) of Wang's paper on finite bases for radical well-mixed difference ideals, where an affirmative answer is conjectured. The conjecture is posed for arbitrary binomial sigma-ideals, whose generators include the general binomials aM1+bM2aM_1+bM_2 with a,b∈Ka,b\in K and M1,M2M_1,M_2 sigma-monomials; an answer either proves the conjecture for arbitrary characteristic-zero difference fields, or exhibits a specific base difference field and binomial sigma-ideal for which ⟨I⟩r\langle I\rangle_r is not binomial. The Laurent case, where all sigma-monomials are invertible, is already settled affirmatively by the support-lattice theory of Gao, Huang and Yuan and is not part of this question.

References

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